nimble (version 0.7.0)

Inverse-Gamma: The Inverse Gamma Distribution

Description

Density, distribution function, quantile function and random generation for the inverse gamma distribution with rate or scale (mean = scale / (shape - 1)) parameterizations.

Usage

dinvgamma(x, shape, scale = 1, rate = 1/scale, log = FALSE)
rinvgamma(n = 1, shape, scale = 1, rate = 1/scale)
pinvgamma(q, shape, scale = 1, rate = 1/scale, lower.tail = TRUE,
  log.p = FALSE)
qinvgamma(p, shape, scale = 1, rate = 1/scale, lower.tail = TRUE,
  log.p = FALSE)

Arguments

x

vector of values.

shape

vector of shape values, must be positive.

scale

vector of scale values, must be positive.

rate

vector of rate values, must be positive.

log

logical; if TRUE, probability density is returned on the log scale.

n

number of observations.

q

vector of quantiles.

lower.tail

logical; if TRUE (default) probabilities are \(P[X \le x]\); otherwise, \(P[X > x]\).

log.p

logical; if TRUE, probabilities p are given by user as log(p).

p

vector of probabilities.

Value

dinvgamma gives the density, pinvgamma gives the distribution function, qinvgamma gives the quantile function, and rinvgamma generates random deviates.

Details

The inverse gamma distribution with parameters shape \(=\alpha\) and scale \(=\sigma\) has density $$ f(x)= \frac{s^a}{\Gamma(\alpha)} {x}^{-(\alpha+1)} e^{-\sigma/x}% $$ for \(x \ge 0\), \(\alpha > 0\) and \(\sigma > 0\). (Here \(\Gamma(\alpha)\) is the function implemented by R's gamma() and defined in its help.

The mean and variance are \(E(X) = \frac{\sigma}{\alpha}-1\) and \(Var(X) = \frac{\sigma^2}{(\alpha-1)^2 (\alpha-2)}\), with the mean defined only for \(\alpha > 1\) and the variance only for \(\alpha > 2\).

See Gelman et al., Appendix A or the BUGS manual for mathematical details.

References

Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. (2004) Bayesian Data Analysis, 2nd ed. Chapman and Hall/CRC.

See Also

Distributions for other standard distributions

Examples

# NOT RUN {
x <- rinvgamma(50, shape = 1, scale = 3)
dinvgamma(x, shape = 1, scale = 3)
# }